Optimal. Leaf size=216 \[ \frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4} \]
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Rubi [A]
time = 0.15, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {748, 828, 857,
634, 212, 738} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}-\frac {(2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^4}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 738
Rule 748
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx &=\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\int \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\int \frac {-\frac {1}{2} b d \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-\frac {1}{2} (2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^3}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\left (d^2 (c d-b e)^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c e^4}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\left (2 d^2 (c d-b e)^2\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c e^4}\\ &=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 238, normalized size = 1.10 \begin {gather*} \frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-48 c^{3/2} d^{3/2} (-c d+b e)^{3/2} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )+3 \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{24 c^{3/2} e^4 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs.
\(2(188)=376\).
time = 0.51, size = 549, normalized size = 2.54
method | result | size |
default | \(\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}}{e}\) | \(549\) |
risch | \(\frac {\left (8 c^{2} e^{2} x^{2}+14 b c \,e^{2} x -12 c^{2} d e x +3 b^{2} e^{2}-30 b c d e +24 d^{2} c^{2}\right ) x \left (c x +b \right )}{24 c \,e^{3} \sqrt {x \left (c x +b \right )}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b^{3}}{16 e \,c^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b^{2} d}{8 e^{2} \sqrt {c}}+\frac {3 \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b \,d^{2}}{2 e^{3}}-\frac {c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{3}}{e^{4}}-\frac {d^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) b^{2}}{e^{3} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {2 c \,d^{3} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) b}{e^{4} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {c^{2} d^{4} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\) | \(627\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.95, size = 879, normalized size = 4.07 \begin {gather*} \left [\frac {{\left (3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + 2 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}\right )} e^{\left (-4\right )}}{48 \, c^{2}}, \frac {{\left (96 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}\right )} e^{\left (-4\right )}}{48 \, c^{2}}, \frac {{\left (3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - 24 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}\right )} e^{\left (-4\right )}}{24 \, c^{2}}, \frac {{\left (48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}\right )} e^{\left (-4\right )}}{24 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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